Introduction to string theory

76 Pages·2004·0.448 MB·English

by  Van Proeyen A.

Institute for Intensive Theoretical Studies
April, 5–16, 2004
Introduction to String Theory
Antoine Van Proeyen †
Institute for theoretical Physics
Celestijnenlaan 200D, B-3001 Leuven
Abstract
This course is aimed at beginning Ph.D. students that are not familiar with elementary particle
physics. The basic concepts of string theory are introduced and its main features are exposed. The
main ingredients of string theory, as extra dimensions, supersymmetry, dualities and branes are
explained and it is shown what is their role in establishing a theory of the particles and all known
forces between them. This should lead to an understanding of the status of research in establishing
a standard model including gravity and a theory of the early universe.
† [email protected];
homepage: http://itf.fys.kuleuven.ac.be/~toine/home.htm
Contents
1 Elementary particle physics 4
1.1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Quantum field theory and the path integral . . . . . . . . . . . . . . . . . . 4
1.3 Elementary interactions and their theories . . . . . . . . . . . . . . . . . . . 6
2 String solutions and states 10
2.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Solutions using boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Physical string modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Open string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Orientable or non-orientable strings . . . . . . . . . . . . . . . . . . . . . . . 17
2.6 Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Strings in perturbation theory 19
3.1 Conformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Amplitudes from surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Example of the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Modular transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Vertex operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6 Effective actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Strings in curved backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Coupling constants and frames . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.9 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 Superstring theories 32
4.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 R and NS boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Physical string modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.1 Open string modes and the GSO projection . . . . . . . . . . . . . . 36
4.4.2 Closed string modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5 Five superstring theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.1 Open and unoriented strings . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.2 Heterotic string theories . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5.3 Summary of the 5 theories . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Dualities 43
5.1 Supergravities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.1 The list of supergravity theories. . . . . . . . . . . . . . . . . . . . . 43
5.1.2 Dimensional reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2
5.2 T-duality for closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.1 Exchanging winding and momenta modes . . . . . . . . . . . . . . . 45
5.2.2 Dimensional reduction in spacetime of the NS-NS sector . . . . . . . 47
5.2.3 T-duality for the superstrings . . . . . . . . . . . . . . . . . . . . . . 48
5.3 T-duality for open strings: D-branes . . . . . . . . . . . . . . . . . . . . . . 48
5.3.1 T-duality induces D8 branes . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.2 Charged strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3.3 Dp-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4 The full T-duality procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5 S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.5.1 Self-duality in IIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5.2 type I - SO(32) heterotic . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 M-theory and Branes 56
6.1 Supersymmetry algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 BPS branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2.1 BPS Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.2.2 Brane solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2.3 Tensions and charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2.4 Dirac charge quantization . . . . . . . . . . . . . . . . . . . . . . . . 63
6.3 The 11th dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.3.1 Strong coupling of IIA superstrings . . . . . . . . . . . . . . . . . . . 65
6.3.2 M-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4 The theory currently known as M . . . . . . . . . . . . . . . . . . . . . . . . 67
7 Selected topics of string theory 69
7.1 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Calabi-Yau manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 Black hole entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.4 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.5 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.6 Some final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A Notations 75
B Resources for popular introductions, these lectures and further study 75
3
1 Elementary particle physics
What are the goals of elementary particle physics?
Which are the established main facts?
What is a particle?
Which are the main theories?
Why do we need string theory?
Goal of high-energy physics: particles and forces.
Use ‘The official string theory web site’ for the standard model.
Remark on pentaquarks: see Physicsweb news.
1.1 Particles
The spacetime is a fixed background on which particles move. The Poincar´e group plays
an important role. An elementary particle is a set of states transforming in an irreducible
representation of the Poincar´e group: the Lorentz rotations SO(1,D −1) and translations.
An irreducible representation is characterized by Casimirs: mass and spin. The mass is
determined by M2 = −p2.
States with M2 < 0 are called tachyons. These are non-stable states: their kinetic energy
decreasesbyincreasingspeed. I.e. theysignalthatwehavenottakentherightvacuum.
We are expanding around a maximal of the potential, rather than a minimum.
For massless particles. Using a Lorentz transformation, we can put the momentum in a
light-like direction e.g. pµ = p+δµ. The states form a representation of SO(D − 2).
+
E.g. in 4 dimensions you can have just one state, or, if there is also parity invariance,
a massless particle of spin s has two helicity states h = ±s.
For massive particles we can put the momentum in the timelike direction (rest frame).
The states form a representation of SO(D−1) (or rather its covering group, meaning
that spinor representations are possible). In 4 dimensions, this is characterized by the
spin s. A particle of spin s has (2s+1) states with helicity h = s, s−1,...,−s.
Remark that we have started here with a ‘flat space’ such that the Poincar´e group is the
main tool. If spacetime is curved, e.g. a (anti-)de Sitter space, the notion of a particle needs
to be reconsidered, but the same methods are used to define it.
1.2 Quantum field theory and the path integral
Theories to describe the particles and forces: quantum mechanics and special relativity,
unified in quantum field theory. The theory is defined by an action, that determines classical
field equations. The quadratic terms in the action determine the propagation of the particles
and the other terms determine their interactions. Often they are multiplied by a coupling
4
constant, e.g. g, such that for weak coupling g (cid:28) 1 one can work perturbatively in this
coupling constant. One can represent the interactions as Feynman graphs, where tree graphs
determine the classical scattering amplitudes, and loops determine the quantum effects.
A convenient way to describe field theory amplitudes uses the Feynman path integral.
In path integral quantization, amplitudes are given by summing over all possible histories
interpolating between the initial and final states. Each history is weighted by
(cid:18) (cid:19) Z
i
exp S , S = dxDL(Φ(x),∂Φ(x)), (1.1)
~ cl cl
where S is the classical action for the given history. The path integral is the quantity
cl
Z
Z = DΦe~iScl, (1.2)
where the integral is taken over all configurations of the fields Φ. Expectation values of
functionals of fields A(Φ) are determined as
R DΦA(Φ)e~iScl
hAi = . (1.3)
R DΦe~iScl
One usually first tries to interpret this definition in perturbation theory in Planck’s con-
stant ~. In a certain regime, the amplitudes can have a meaningful, although most likely
asymptotic, expansion of the form
X
hAi ∼ A ~g, (1.4)
g
n
where the perturbative coefficients A are computed as sums over all Feynman diagrams Γ
g
with a fixed number of g loops
X w
Γ
A = . (1.5)
g
#Aut(Γ)
Γ
Here the individual weight w of the diagram Γ is computed using the Feynman rules and
Γ
we divide by the order of the symmetry group of the diagram. This relation with graphs
explains why quantum field theories can describe point-particles and their interactions.
Loops involve an integral over momentum that can circulate. Lines in the loop are not
‘on-shell’, i.e. p pµ 6= −m2. These integrals may diverge. This is solved in most theo-
µ
ries by regularization and renormalization. Regularization is the process of giving meaning
to the infinite expressions. A parameter is defined such that the integral only diverges if
that parameter obtains a critical value. Regularization means that terms are added to the
action that depend on this parameter, and that also diverge if this parameter obtains the
critical value. But this is arranged in such a way that the sum of the original and new
Feynman graphs give expressions such that one obtains a finite result. Though this seems a
strange procedure, the results for most theories are surprisingly accurate in agreement with
experiments. However, for quantum gravity this does not work.
5
Finally, I want to draw the attention to ‘anomalies’, which are another aspects of quan-
tization of gauge theories. Often the quantization breaks gauge symmetries. Technically,
this comes about because there is no regularization scheme that respects the symmetry (the
symmetry is only valid when the regularization parameter is equal to its critical value). An
anomaly in a rigid symmetry is often a physical feature that can be measured, and can be
found in agreement with an experiment. However, an anomaly in a local symmetry means
that some degrees of freedom appear that were not present in the classical theory. This
might lead to inconsistencies. Often the absence of such anomalies gives a restriction on
the field content of consistent theories. It might give an equation on the number of certain
matter representations in the gauge group.
Exercise 1.1: (Advanced). What are the anomaly restrictions on the standard
model?
How do they come about?
1.3 Elementary interactions and their theories
Electromagnetism is mediated through a vector field A that describes a spin-1 particle. A
µ µ
has D components, while a spin-1 particle is a representation of SO(D−2) and carries D−2
physical degrees of freedom. Two degrees of freedom are eliminated from the 4 components
of A by the gauge invariance δA = ∂ Λ and the field equation eliminates then one more
µ µ µ
degree of freedom. One may eliminate these degrees of freedom by explicit ‘gauge choices’,
determining some components of A . Often it is more useful to keep the Lorentz invariance,
µ
and describing the theory using ‘Faddeev-Popov ghost’ fields. These are non-physical fields
that ‘compensate’ in some way the gauge degrees of freedom of other fields in the theory.
E.g. in this case, the ghost corresponding to the Λ transformation and a corresponding
antighost reduce the effective physical degrees of freedom of the gauge field to D−2.
Chargedfields, liketheelectron,arefieldsthattransformunderthegaugetransformation.
For electromagnetism this transformation is a multiplication by a complex phase factor, i.e.
ψ → eieqΛψ where e is a the unit charge and q is a real number, determining the charge of the
particle as qe. For later reference: δψ = ieqΛψ. They feel the electromagnetism because in a
gauge-invariance lagrangian, their kinetic term should be in the form D ψ = ∂ ψ −ieqA ψ
µ µ µ
in order that this does not transform with derivatives on the parameter. In this case, the
gauge transformation was just a U(1) group.
Weak interactions are described together with electromagnetism (electro-weak unifica-
tion) through gauge symmetry with an SU(2) × U(1) gauge group. This involves 4 gen-
erators and in a gauge theory we then need 4 gauge fields. Gauge fields are fields whose
transformation involves ∂ ΛI where ΛI are the parameters. ‘Matter fields’ are fields that
µ
are in a representation of the gauge group. They feel the interaction if this representation
is non-trivial, again though the appearance of covariant derivatives in the action. Another
important part of the action comes into play here: the scalar potential V(φ), where φ stands
for the scalar fields. It is the part of the Lagrangian that depends only on the scalars and
does not contain derivatives. Its minimum determines vacuum states where the other fields
6
are zero (preserving Lorentz invariance) and the scalar fields are constants. Such vacua may
not be invariant under the symmetry group. In that case we say that the gauge symmetry
is spontaneously broken. The gauge fields corresponding to the broken symmetries are mas-
sive. This is what happens in the electroweak theory. The vacuum is only invariant under
a linear combination of the U(1) factor in the gauge groups and a generator that belongs
to SU(2). This linear combination corresponds to the gauge symmetry of electromagnetism.
The other 3 gauge fields are the ‘W and Z massive gauge bosons’.
Strong interactions are described by a gauge theory of SU(3), that acts on the colours
of the quarks. Quarks are in triplet representations, and the gauge fields are the gluons. A
main aspect of strong interactions is that they keep the quarks confined within mesons and
hadrons. It is not proven that the theory of SU(3) performs this binding. The problem is
that this is an effect due to strong coupling. Hence, it can not be proven by perturbation
theory. We need a non-perturbative analysis of the theory to obtain such a result. Strong
indications have been given by lattice gauge theories. This is an approximation where the
continuousspacetimeisreplacedbyalatticeandstrongcomputermethodsareusedtoderive
properties of the particles. String theory will indicate a new way, as we will see below.
Thepreviouspartstogether, atheorywithgaugegroupSU(3)×SU(2)×U(1)iscalled‘the
standard model’. Itthuscontainsthegaugefieldsandthe‘mattermultiplets’,representations
of the group whose components are the leptons and the quarks.
Exercise 1.2: Which are the full set of representations that are included to build the
presently known standard model? What are the modifications due to the recent
observations of massive neutrinos?
Extensions have been made ‘unifying’ SU(3)×SU(2)×U(1) in one using simple groups
like SU(5) or E . These have more generators, hence more forces. Some of these forces may
6
mediate the decay of the proton. To limit this decay in the observational limits is one of
the main problems. This approach is called ‘grand unification’. The success was limited and
moreover, it still neglects one main force: gravity.
Exercise 1.3: We mention here for the first time an exceptional group: E . Do
6
you know the catalogue of all simple Lie groups (or Lie algebras)? Make a
table including their ranks and their dimensions. Consider also the accidental
degeneracies for groups of low rank.
The theory of gravity is general relativity, which makes use of a curved spacetime. In
˜
field theory as mediated by a spin 2 particle g = η +κζ . Concerning units: the Newton
µν µν µν
constant is defined as G in
N
G m m
N 1 2
F = , (1.6)
12 |r |D−2
12
where for later convenience, I write D − 2 rather than 2 to be able to generalize to more
dimensions. For D = 4 we have G = 6.7×10−11m3kg−1s−2. The Planck mass M2−D = G
N Planck N
and length ‘ = M−1 are then defined using the ~ = c = 1 conventions, and are in 4
Planck Planck
dimensions
M = 2.2×10−8kg = 1.22×1019GeV , ‘ = 1.6×10−35m (1.7)
Planck Planck
7
We use GeV as unit for energy which is 1.6×10−10J or, dividing by c2, is equivalent with
√
a mass of 1.8×10−27kg, or 5×1015m−1. We often use also in D = 4 M ≡ M / 8π ∼
P Planck
2.4 × 1018 GeV and κ = M−1 and in general dimensions M ≡ M (8π)1/(2−D) and
√ P P Planck
κ ≡ M(2−D)/2 = 8πG .
P N
The action for general relativity (GR) is then
Z
1 √
S = dDx gR(g), (1.8)
GR 2κ2
where R(g) is the scalar curvature in the conventions that compact spaces have positive
curvature.
Gravity can be seen as the gauge theory of the symmetry of general coordinate transfor-
mations. The metric is then the gauge field for these gauge transformations.
Exercise 1.4: (repetition of elements of GR): Which are the building stones to con-
struct an invariant action in GR? E.g. when we use εµνρσ, do we still need a
√
g ? Is the value of this object a number ±1 consistent with general coordinate
transformations? Can we raise and lower its indices. How do we construct invari-
ant actions when fermions are present? What is the relation between the spin
connection and the Levi-Civita connection? What is the relation between the
curvature tensors defined using these two connections? What is the definition
of the scalar curvature mentioned above? Check the signs, and verify the last
statement for a 2-sphere.
Suppose that I modify the metric with a scale factor g0 = φg . What is then
µν µν
the modification of the action (1.8)?
Other attempts have been made to unify gravity with the other interactions in a quantum
theory. A noteworthy attempt is supersymmetry and supergravity. While in the previously
mentioned gauge theories, bosons and fermions are always in separate representations of
the gauge group, here bosons and fermions are necessary to form a representation of the
invariance group.
Exercise 1.5: Prove that the action
Z
S = d4x(cid:2)−1∂ A∂µA− 1∂ B∂µB − 1ζ¯∂/ζ(cid:3), (1.9)
2 µ 2 µ 2
where ζ is a Majorana (‘real’) spinor, is invariant under the transformations
δA = (cid:15)¯ζ,
δB = i(cid:15)¯γ ζ,
5
δζ = ∂/(A+iγ B)(cid:15), (1.10)
5
where (cid:15) is the parameter of the transformation (rigid, i.e. not x-dependent).
What you need about Majorana spinors in 4 dimensions is that for two of these
¯ ¯ ¯
λχ = χ¯λ, λγ χ = χ¯γ λ, λγ χ = −χ¯γ λ. (1.11)
5 5 µ µ
8
where λ¯ ≡ λTC with an antisymmetric ‘charge conjugation matrix’ C. The alge-
bra of γ-matrices implies that ∂/∂/ = (cid:3).
The commutator of two supersymmetry transformations gives (after use of field equations)
a translation.
When supersymmetry is promoted to a gauge symmetry, then this involves automatically
local translations, i.e. general coordinate transformations. Therefore it involves GR. As
mentioned, the gauge field of the general coordinate transformations is the metric, a spin-2
field. The gauge fields of supersymmetry is a spin-3/2 field, which is called the gravitino.
The supersymmetry restricts the quantum corrections produced by loops. This occurs
because loops of fermions in field theory have a minus sign with respect to loops of bosons.
Supersymmetry arranges it such that many divergences are cancelled. It was for some time
hoped that supergravity would be a finite theory of gravity. However, the divergences still
appear, although only for a large number of loops (3 loops in the simplest supergravity).
There are extensions possible with more supersymmetry generators (more such parameters).
However, there are limits. The highest extensions has 8 such spinor parameters (cid:15)i, with
i = 1,...,8. However, that still does not produce full finite results, and it has not enough
freedom to include the standard model.
Exercise 1.6: What is the minimal supergravity action? Can you prove that it is
invariant?
So we are left with the question: how can nature be described, taking into account that
gravity exists.
Video: chapter 6 and 7 of hour 1: the problems of two types of theories and the idea of
strings (12’).
Video: part 1 and 2 of 2nd hour (somewhat repetition of previous parts) (13’).
9
2 String solutions and states
How are string vibrations described?
Do, or how do, strings end?
What are the masses of the string states?
Why do strings live in more than 4 spacetime dimensions?
A string along the x direction, vibrating in y direction has the wave equation
∂2y(x,t) ∂2y(x,t)
= v2 , (2.1)
∂t2 ∂x2
where v is the wave velocity. Solutions for a string of length L are
∞ (cid:18) (cid:19)
X nπvt nπvt nπx
y(x,t) = a cos +b sin sin . (2.2)
n n
L L L
n=1
The frequency of the n-th mode is
nv
f = . (2.3)
n
2L
In a relativistic theory there is not such a clear difference between time and space and we
have to write covariant equations. The string sweeps out a 2-dimensional surface with (τ,σ)
and the equation is
δ2Xµ(σ,τ) δ2Xµ(σ,τ)
= c2 . (2.4)
∂τ2 ∂σ2
Solution:
√ X±∞ 1 nπσ
Xµ(σ,τ) = xµ +2α0pµτ +i 2α0 αµe−inπcτ/Lcos . (2.5)
n n L
n6=0
with (α )∗ = α . This is an open string with ends that are floppy.
n −n
2.1 Actions
Can be obtained from requirement of minimal surface. Element of surface is σα = (τ,σ)
∂Xµ∂Xν
dσµν = εαβdσdτ (2.6)
∂σα ∂σβ
The Nambu-Goto (NG) action is proportional to the surface
q
S ∝ 1|dσµνdσ |. (2.7)
NG 2 µν
We find
Z
1
S = − d2σ|detG |1/2 . (2.8)
NG 2πα0 αβ
Σ
10